R/02_fof_pc.R
fof_pc.Rd
Function-on-function linear regression based on principal components. This function performs multivariate functional principal component analysis (MFPCA) to extract multivariate functional principal components from the multivariate functional covariates as well as from the functional response, then it builds a linear regression model of the response scores on the covariate scores. Both functional covariates and response are standardized before the regression. See Centofanti et al. (2021) for additional details.
fof_pc(
mfdobj_y,
mfdobj_x,
tot_variance_explained_x = 0.95,
tot_variance_explained_y = 0.95,
tot_variance_explained_res = 0.95,
components_x = NULL,
components_y = NULL,
type_residuals = "standard"
)
A multivariate functional data object of class mfd denoting the functional response variable. Although it is a multivariate functional data object, it must have only one functional variable.
A multivariate functional data object of class mfd denoting the functional covariates.
The minimum fraction of variance that has to be explained by the multivariate functional principal components retained into the MFPCA model fitted on the functional covariates. Default is 0.95.
The minimum fraction of variance that has to be explained by the multivariate functional principal components retained into the MFPCA model fitted on the functional response. Default is 0.95.
The minimum fraction of variance that has to be explained by the multivariate functional principal components retained into the MFPCA model fitted on the functional residuals of the functional regression model. Default is 0.95.
A vector of integers with the components over which
to project the functional covariates.
If NULL, the first components that explain a minimum fraction of variance
equal to tot_variance_explained_x
is selected.
#' If this is not NULL, the criteria to select components are ignored.
Default is NULL.
A vector of integers with the components over which
to project the functional response.
If NULL, the first components that explain a minimum fraction of variance
equal to tot_variance_explained_y
is selected.
#' If this is not NULL, the criteria to select components are ignored.
Default is NULL.
A character value that can be "standard" or "studentized". If "standard", the MFPCA on functional residuals is calculated on the standardized covariates and response. If "studentized", the MFPCA on studentized version of the functional residuals is calculated on the non-standardized covariates and response. See Centofanti et al. (2021) for additional details.
A list containing the following arguments:
* mod
: an object of class lm
that is a linear regression model
where
the response variables are the MFPCA scores of the response variable and
the covariates are the MFPCA scores of the functional covariates.
mod$coefficients
contains the matrix of coefficients
of the functional regression basis functions,
* beta_fd
: a bifd
object containing the
bivariate functional regression coefficients \(\beta(s,t)\)
estimated with the function-on-function linear regression model,
* fitted.values
: a multivariate functional data object of
class mfd with the fitted values of the
functional response observations based on the
function-on-function linear regression model,
* residuals_original_scale
: a multivariate functional data object
of class mfd
with the functional residuals of the
function-on-function linear regression model on the original scale,
i.e. they are the difference between
mfdobj_y
and fitted.values
,
* residuals
: a multivariate functional data object of class mfd
with the functional residuals of the
function-on-function linear regression model,
standardized or studentized depending on
the argument type_residuals
,
* type_residuals
: the same as the provided argument,
* pca_x
: an object of class pca_mfd
obtained by doing MFPCA on the functional covariates,
* pca_y
: an object of class pca_mfd
obtained by doing MFPCA on the functional response,
* pca_res
: an object of class pca_mfd
obtained by doing MFPCA on the functional residuals,
* components_x
: a vector of integers
with the components selected in the pca_x
model,
* components_y
: a vector of integers
with the components selected in the pca_y
model,
* components_res
: a vector of integers
with the components selected in the pca_res
model,
* y_standardized
: the standardized functional response
obtained doing scale_mfd(mfdobj_y)
,
* tot_variance_explained_x
: the same as the provided argument
* tot_variance_explained_y
: the same as the provided argument
* tot_variance_explained_res
: the same as the provided argument
* get_studentized_residuals
: a function that allows
to calculate studentized residuals on new data,
given the estimated function-on-function linear regression model.
Centofanti F, Lepore A, Menafoglio A, Palumbo B, Vantini S. (2021) Functional Regression Control Chart. Technometrics, 63(3), 281--294. <doi:10.1080/00401706.2020.1753581>